Von Neumann and Newman poker with a flip of hand values

Early Round Bluffing in Poker

“…We use the notation p 1 (b, [0, c]) to represent the probability of Player 1 winning, conditional on h 1 = b and h 2 ∈ [0, c]. With this notation, equations (1) become 1], c))) = 0. (6) We'll come back to these equations, or their equivalent using the p y notation introduced below, throughout the paper.…”

“…The net effect is that relative hand value changes over the course of the game. An elegant paper by Bernasconi et al [1] addresses volatility by adding a step in which the hand values are reversed some of the time. We'll discuss the Bernasconi paper further in Section 8.…”

“…If, on the other hand, you think your opponent is bluffing more than 6.7% of the time (100% − 93.3%), you should call more than the equilibrium amount. 1 For calculations such as this, where r = R, you need various p y integrations , shown in Table A.1 of the Appendix. The procedure is this: choose a value of r , and use a solver to combine the appropriate p y expression with the indifference equations (6).…”

“…Both players start with a fixed hand of 0, and so both should have an equal chance of winning. The equation evaluates to 1 2 .…”

Symmetric Form of the Von Neumann Poker Model

“…The model is applicable to river (last card) situations in Texas Holdem, Seven Card Stud and Omaha [11], where hand rankings follow the Well Ordering Principle. Pre-flop play most poker games may be modeled using flip [1]. Note that the symmetric von Neumann model of Goldman and Stone [6] is different in that their model requires the players to act simultaneously, thereby eliminating positional advantage, and players may "drop" or fold rather than checking.…”